Talk:Eigenstate thermalization hypothesis

This article was accepted after consultation with editors at WP:Wikiproject Physics.

Article Overhaul
Hello,

As you can see, I've made a very large contribution/alteration to this article. I was initially under the impression that no such article existed, since this article is currently an orphan (with no other articles linking to it), and the article "Eigenstate Thermalization Hypothesis," with a different capitalization, was listed as a requested article.

Since I am one of Mark Srednicki's graduate students, and I have been doing research in this particular area, I figured I would put together a comprehensive discussion of the subject. I tend to be somewhat obsessive about my work, so the article was relatively detailed before it went live. However, it was ultimately brought to my attention that the article already existed, and so my submission was rejected, with the advice that I merge my work into this article.

Because of the disparity in length and level of detail in the two articles, it seemed to be an incredibly difficult task for me to simply merge my article into the existing one without necessarily making large changes. Having some knowledge of this subject, I found that many of the sentiments in the original article were still reflected in my version, only worded in a slightly more detailed manner. I also took specific issue with certain statements that I believed to be incorrect, and I made sure to put in proper references to the appropriate literature to support the opposing claims.

I assume that there may be some contention over this edit, since large portions of the original article were reworded. I please ask that if anyone takes issue with sentences or statements that were removed from the original article, that instead of reverting my entire change, they simply add those specific statements back into the article with a subsequent edit. I feel as though my edit has contributed a large amount of new information which was not previously contained in the article, and serves to better the article. KeithFratus (talk) 21:06, 24 September 2013 (UTC)


 * Thanks very much for the major rewrite and expansion of this article. It will take a while to digest, but a first glance the prose looks great. I am a little concerned, with regard to keeping a neutral point of view in the article, that all the refs about ETH in particular seem to be from Srednicki's group and close collaborators. Are there other groups doing research in this area? If so it may be prudent to include their take, along with the most relevant paper(s) and/or reviews. For instance, there is a paper on the eigenstate randomizations hypothesis that looks like a generalization of ETH. I know of the review, but as far as I know it has not been published yet, so does not count as a reliable source. Thanks, --Mark viking (talk) 22:14, 24 September 2013 (UTC)


 * Yes, this is a good point which I want to eventually address. There are a lot of related ideas similar to ETH, and other people who are working on the subject, but I haven't yet had the time to add all of the references. In order to get the article up and running, I put in the minimal number of references which seemed sufficient for the time being, but eventually I want to greatly expand the number of references. There is also the related subject of Many-Body Localization, which seems, from what I understand, to effectively be the opposite situation of ETH, along with various possible extensions of the ETH to more general situations (for example, understanding how it might explain the validity of a grand canonical ensemble, instead of just the usual canonical ensemble). There are also some independent researchers who have attempted to understand the ETH in terms of the Central Limit Theorem and Chaotic Eigenstates (Srednicki himself seemed to be unaware of this work before I showed it to him personally last week).
 * I've seen the Eigenstate Randomization paper before, but I haven't had time to read it carefully yet. Part of the reason I wanted to get this article up here is because it seems there have been a lot of recent papers in which people are incorrectly using the term "Eigenstate Thermalization Hypothesis," stating something which is actually different from what Mark Srednicki and others have been describing. There are some papers which claim to disprove the ETH in some cases, but many of these authors seem to have a very different notion of what "thermalization" means, and allow for an explicit averaging over initial states, which sort of defeats the purpose of ETH's original question.
 * Anyways, I'm currently in the process of organizing a collection of all of the relevant literature for my own use, so as soon as I find some time, I'll try to add more references to this article. Anyways, thanks for your input! KeithFratus (talk) 01:47, 25 September 2013 (UTC)

Orphan Status
I've created a link to this article from Statistical_mechanics. There was already a discussion of Berry's conjecture, so it seemed natural to include a link to the ETH. I'm sure there are other places where such a link would be justified. Should the orphan status be removed now? KeithFratus (talk) 21:35, 24 September 2013 (UTC)


 * I have removed the tag. --Mark viking (talk) 22:16, 24 September 2013 (UTC)

Points of Contention in the Original Article
I would like to extend a thanks to the original author for creating the article on this important subject. However, as someone who has been working carefully with the subject lately, I had a few concerns about some of the content. Here are some points which I was concerned with in the original article.

First, there is the statement:


 * "Clearly there must be a difference between quantities that are experimentally observable, which are often referred to as “simple” or “local”observables, and the eigenstate amplitudes."

This is actually not correct, which is pointed out in the new section "General Validity of the ETH," and is supported by the reference "Thermalization and its mechanism for generic isolated quantum systems." Observables need not be local in order to satisfy the ETH, and there is, to my knowledge, no precise notion (yet) of how "simple" they must be in order to do so.

Secondly, there is the claim:


 * "The eigenstates of a many-body quantum system are extraordinarily complex and their individual amplitudes cannot be measured. The memory that is implicit in the conservation of eigenstate amplitudes is not in itself measurable. But these amplitudes ultimately determine the values of simpler observables."

I am not sure what specific notion of complexity is being referred to here, but this is certainly not true in general many-body systems. For example, integrable systems can be written in terms of number operators which all commute with each other, and the eigenstates of such systems have an incredibly simple representation as product states in Fock space. This is all well-supported by the references I have included. Additionally, the fact that "amplitudes" cannot be measured is not a special feature of many-body systems - this is true for any quantum-mechanical system, since the state vector is never directly observed.

Lastly, there is the statement:


 * "The memory of initial conditions is effectively obliterated when a system thermalizes. "

I am not sure precisely how "obliterated" is to be defined in this context, but I think this statement may be somewhat too general. I would contend that one of the main points of the ETH is that fundamentally, information is NOT lost in the time evolution of a quantum system, since all of the information about the state preparation is contained in the diagonal ensemble. Rather, while information is retained, the ETH allows us to construct a simplified description of the system, in terms of statistical ensembles, which produces negligible errors in the thermodynamic limit (assuming that ETH is correct). This is partly why I have been trying to avoid, as much as possible, the word "thermalizaton" (which is ironic, given the name of the hypothesis). The reason is that, as pointed out by Mark in his talk at the KITP, the precise definition of the word "thermalizaton," as it regards an entire system, tends to be somewhat contentious, and it is better to rather think of the ETH as clarifying when a given statistical ensemble will provide an accurate description of a system, which is a much more precise statement.

Lastly, there were large sections of the original article I agreed with, although I felt as though they could be made much more precise:


 * "For any observable, at long times compared to the inverse of the spread in Bohr frequencies in a system, the off-diagonal matrix elements tend to average to zero. So at long times, the expectation value of an observable is just the weighted average of that observable over all the occupied eigenstates of the system. The ETH is that the expectation values of simple observables are nearly the same for all comparable energy eigenstates of a many-body quantum system and equal to the microcanonical value, that is, to the value obtained by finding the most probable state in a statistical mechanical analysis. If the ETH is satisfied for an observable, then that observable will thermalize. If the ETH is not satisfied for an observable, then it is possible to prepare an initial state of the system for which, in the infinite time limit, the observable does not approach the microcanonical value.The statement “This isolated many-body quantum system thermalizes” is equivalent to the statement “All measurable observables of this many-body quantum system satisfy the ETH”."

These ideas are now addressed in the section on the diagonal ensemble, where I believe I have altered the wording to be more clear.

Again, I thank the original author for creating this article, and I only mean this comments to be constructive criticism. I am happy to discuss possible changes to the article to reflect any sentiments which were contained in the original article. KeithFratus (talk) 21:57, 24 September 2013 (UTC)

More similar to H-theorem than ergodicity, no?
The article tries to draw a comparison to the ergodic hypothesis but I'm not sure if that's the right connection to make.

The statement of the ETH seems to resemble very closely that of the H-theorem. Similarities include --Nanite (talk) 23:56, 2 December 2013 (UTC)
 * The system quickly seeks out an equilibrium-ish value of the observable due to chaotic behaviour.
 * The observable fluctuates. And we can expect quantum revivals (analogous to Poincare recurrence).


 * I think you are probably correct that the ETH is not really directly analogous to the ergodic hypothesis, although I would argue that this is stated in the article. In the article, the ergodic hypothesis is discussed as the reason why statistical mechanics is successful in making predictions in classical mechanics. The article then contrasts the classical case with the quantum case, in which there is no sense of dynamical chaos, and so some alternative explanation is necessary for why quantum statistical mechanics is successful. At least when discussing the validity of the microcanonical ensemble in classical statistical mechanics, discussions of ergodicity should be valid - I can certainly think of a few textbook references which make a point of emphasizing the idea that the ergodic hypothesis is what forms the justification for classical statistical mechanics. But in any event, this is certainly not just my personal opinion - all of the literature I have seen on the ETH seems to use the language of ergodicity when contrasting the classical and quantum cases. In particular, in the 2008 Nature article by Rigol et al, which seems to have started all of the recent interest in ETH, the very first paragraph contains the word "ergodically."


 * Also, while there may be some similarities to the H-theorem, it would seem that the H-theorem is specifically concerned with the time-evolution of the "entropy" of the system, and not more general observables. From my discussions with Mark Srednicki, it seems like the issue of entropy in isolated quantum systems may be somewhat of a contentious point, and I'm not so sure that discussion of the ETH even really requires a well-formed notion of entropy in order to make sense. Also, it seems like the H-theorem is somewhat particular to the kinetic theory of gasses. So I think while it might be related, the ETH concerns issues which are a little bit more general. But it's certainly related enough to put a link to the H-theorem article at the bottom of the page - I'll do that now if it hasn't already been done.


 * Anyways, with that all said, if you can think of specific changes to make the article, I'd be happy to discuss them. Recently my schedule has lightened up a bit and there are plenty of changes I'd like to make to the article to make it a little more well-written. KeithFratus (talk) 03:27, 7 January 2014 (UTC)

"Chaotic" nature of most "realistic" systems
I see the addition of a "dubious" flag in relation to this statement, and I admit that something better should be written. At the time of writing I was a little pressed for time, so I didn't really come up with the best language. Of course, as Mark Srednicki himself points out in one of the talks he gave on the subject, the use of the phrase "most systems" is a little bit subtle, since if we want to decide what exactly we mean by "most systems," we need some sort of metric to quantify the set of all physical systems. Due to this subtlety, I was a little lazy in my original wording. I'll try to think of a better way to phrase this issue. The basic idea I was trying to get at is that most authors on the subject seem to cite dynamical chaos as the "rigorous" justification for using statistical mechanics to describe classical equilibrium systems.KeithFratus (talk) 03:32, 7 January 2014 (UTC)

Update: I changed the wording of this section to be more accurate. I think all of the language is now supported by claims made in the textbook by Reichl, among other resources. KeithFratus (talk) 03:52, 7 January 2014 (UTC)

Linear evolution is not in conflict with classical dynamical chaos
This article makes a false claim that is repeated in the introduction of many papers: that the linear evolution described by the Schrodinger equation is somehow in conflict with ideas from classical dynamical chaos.
 * However, this mechanism of dynamical chaos is absent in Quantum Mechanics, due to the strictly linear time evolution of the Schrödinger equation...This time evolution is manifestly linear, and any notion of dynamical chaos is absent. Thus, it becomes an open question as to whether an isolated quantum mechanical system, prepared in an arbitrary initial state, will approach a state which resembles thermal equilibrium, in which a handful of observables are adequate to make successful predictions about the system.

The problem with this is that it conflates two completely different metric: distance in phase space and distinguishability of states. Classical chaos can be characterized by the exponential divergence of nearby states in phase space, but two classical probability distributions that are difficult to distinguish remain just as distinguishable (as measured, say, by the KL divergence) under chaotic Hamiltonian flow. Likewise, quantum chaos can be characterized by the exponential divergence, in phase space distance, of nearby states (as measured, say, by the difference of their expectation values $$\langle \hat{x}\rangle, \langle \hat{p}\rangle$$, but two quantum states, pure or mixed, remain just as distinguishable (as measured, say, by the quantum analog of the KL divergence, i.e., the relative entropy). Jess (talk) 13:35, 29 April 2017 (UTC)

See for instance, page 347 of A. Peres, Quantum Theory: Concept and Methods (2002). Jess (talk) 22:50, 2 May 2017 (UTC)